arXiv subjects

P. K. Hung

Publications and source records attributed to P. K. Hung.

4 recordsLinked to original sources

arXiv2026

A metric on $S^2 \times S^2$ with positive sectional curvature

We construct a metric on $S^2 \times S^2$ with positive sectional curvature. Starting from the standard metric on $S^2 \times S^2$, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-M\"uter metric. We then consider a suitable third order perturbation of this Cheeger-M\"uter metric and show that the perturbed metrics have positive sectional curvature. The proof requires various calculations, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.

math.DG↗

arXiv2025

The rigidity statement in the Horowitz-Myers conjecture

In this paper, we give an alternative proof of the Horowitz-Myers conjecture in dimension $3 \leq N \leq 7$. Moreover, we show that a metric that achieves equality in the Horowitz-Myers conjecture is locally isometric to a Horowitz-Myers metric.

math.DG↗

arXiv2024

Systolic inequalities and the Horowitz-Myers conjecture

Let $n$ be an integer with $3 \leq n \leq 7$, let $M$ be a compact manifold of dimension $n$ with boundary $\partial M$, and let $g$ be a Riemannian metric on $M$ with scalar curvature at least $-n(n-1)$. Under a topological assumption on $M$, we establish an inequality relating the infimum of the boundary mean curvature to the systole of the boundary $\partial M$. As a consequence, we obtain a new positive energy theorem, with equality being attained by the Horowitz-Myers metrics.

math.DG↗

arXiv2017

Area bounds for minimal surfaces that pass through a prescribed point in a ball

Let $Σ$ be a $k$-dimensional minimal submanifold in the $n$-dimensional unit ball $B^n$ which passes through a point $y \in B^n$ and satisfies $\partial Σ\subset \partial B^n$. We show that the $k$-dimensional area of $Σ$ is bounded from below by $|B^k| \, (1-|y|^2)^{\frac{k}{2}}$. This settles a question left open by the work of Alexander and Osserman in 1973.

math.DG↗